Is it permissible to take 20% of the monthly profit in exchange for investing 20% of the value of my father's car, which he uses as a driver, and is this considered usury (riba)?

It is permissible for you to be a partner with your father in car ownership if you pay one-fifth of its price, for your father to work on it, and for the profit to be shared between you according to the agreement. If your share of the profit is 20%, this means that your father did not take anything in return for his work on your share. If he is donating his work, then there is no harm. But if he is not donating, then your share should be less than 20% so that your father has compensation for his work.